A Wave Equation-Based Hybridizable Discontinuous Galerkin–Robin Transmission Condition Algorithm for Electromagnetic Problems Analyzing

Xuan Zhang; Shi Min Liu; Ran Zhao; Xiao Chun Li; Jun Fa Mao; Li Jun Jiang; Ping Li

doi:10.1109/TMTT.2022.3225324

A Wave Equation-Based Hybridizable Discontinuous Galerkin–Robin Transmission Condition Algorithm for Electromagnetic Problems Analyzing

Xuan Zhang, Shi Min Liu, Ran Zhao, Xiao Chun Li, Jun Fa Mao, Li Jun Jiang, Ping Li

Computer, Electrical and Mathematical Sciences and Engineering

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this work, a wave-equation-based discontinuous Galerkin (DG) method hybridized with the Robin transmission condition (DG-RTC) is developed to solve the frequency-domain electromagnetic (EM) problems. The proposed DG method directly discretizes the vector electric field wave equation in each subdomain, and subsequently, a term named numerical flux is introduced at the subdomain interfaces to connect the solutions between neighboring subdomains. However, the numerical flux depends not only on the electric field E but also on the magnetic field H residing over the interface. Thereby, another equation is essential for solving E and H simultaneously. Realizing that H only situates at the interface of adjacent subdomains, it is thus desired that the auxiliary equation also merely involves H at the interfaces. To reach this aim, at the subdomain interfaces, a Robin transmission condition (RTC) based on the tangential continuity of EM fields across the interface is introduced as the second equation. With this proposed DG-RTC algorithm, only E is a volume variable, while H is a two-dimensional (2-D) one, and the degrees of freedom (DoFs) are thus greatly reduced compared with the traditional DG method. On the other hand, in the frequency domain, the established subdomain matrix equations are implicitly coupled with each other, resulting in a global matrix system. Directly solving it is not yet cheap. To alleviate the computational cost, a finite-element tearing and interconnecting (FETI)-like approach resorts. With the block-diagonal preconditioner and a restriction operator, the global matrix is decomposed into a surface matrix equation and several local matrix equations pertinent to each subdomain. In this way, a direct solver can be applied to solve these small matrix equations efficiently. To validate the proposed DG method in solving various EM problems, several representative examples are presented.

Original language	English (US)
Pages (from-to)	1-11
Number of pages	11
Journal	IEEE Transactions on Microwave Theory and Techniques
DOIs	https://doi.org/10.1109/TMTT.2022.3225324
State	Published - Dec 8 2022

Bibliographical note

KAUST Repository Item: Exported on 2022-12-13
Acknowledgements: This work was supported in part by the Shanghai Committee of Science and Technology under Grant 20501130500 and in part by the NSFC under Grant 62071290 and Grant 61771311

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10.1109/TMTT.2022.3225324

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@article{56bbb987b77c4c46a7be5186fb1fd0df,

title = "A Wave Equation-Based Hybridizable Discontinuous Galerkin–Robin Transmission Condition Algorithm for Electromagnetic Problems Analyzing",

abstract = "In this work, a wave-equation-based discontinuous Galerkin (DG) method hybridized with the Robin transmission condition (DG-RTC) is developed to solve the frequency-domain electromagnetic (EM) problems. The proposed DG method directly discretizes the vector electric field wave equation in each subdomain, and subsequently, a term named numerical flux is introduced at the subdomain interfaces to connect the solutions between neighboring subdomains. However, the numerical flux depends not only on the electric field E but also on the magnetic field H residing over the interface. Thereby, another equation is essential for solving E and H simultaneously. Realizing that H only situates at the interface of adjacent subdomains, it is thus desired that the auxiliary equation also merely involves H at the interfaces. To reach this aim, at the subdomain interfaces, a Robin transmission condition (RTC) based on the tangential continuity of EM fields across the interface is introduced as the second equation. With this proposed DG-RTC algorithm, only E is a volume variable, while H is a two-dimensional (2-D) one, and the degrees of freedom (DoFs) are thus greatly reduced compared with the traditional DG method. On the other hand, in the frequency domain, the established subdomain matrix equations are implicitly coupled with each other, resulting in a global matrix system. Directly solving it is not yet cheap. To alleviate the computational cost, a finite-element tearing and interconnecting (FETI)-like approach resorts. With the block-diagonal preconditioner and a restriction operator, the global matrix is decomposed into a surface matrix equation and several local matrix equations pertinent to each subdomain. In this way, a direct solver can be applied to solve these small matrix equations efficiently. To validate the proposed DG method in solving various EM problems, several representative examples are presented.",

author = "Xuan Zhang and Liu, {Shi Min} and Ran Zhao and Li, {Xiao Chun} and Mao, {Jun Fa} and Jiang, {Li Jun} and Ping Li",

note = "KAUST Repository Item: Exported on 2022-12-13 Acknowledgements: This work was supported in part by the Shanghai Committee of Science and Technology under Grant 20501130500 and in part by the NSFC under Grant 62071290 and Grant 61771311",

year = "2022",

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doi = "10.1109/TMTT.2022.3225324",

language = "English (US)",

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T1 - A Wave Equation-Based Hybridizable Discontinuous Galerkin–Robin Transmission Condition Algorithm for Electromagnetic Problems Analyzing

AU - Zhang, Xuan

AU - Liu, Shi Min

AU - Zhao, Ran

AU - Li, Xiao Chun

AU - Mao, Jun Fa

AU - Jiang, Li Jun

AU - Li, Ping

N1 - KAUST Repository Item: Exported on 2022-12-13 Acknowledgements: This work was supported in part by the Shanghai Committee of Science and Technology under Grant 20501130500 and in part by the NSFC under Grant 62071290 and Grant 61771311

PY - 2022/12/8

Y1 - 2022/12/8

N2 - In this work, a wave-equation-based discontinuous Galerkin (DG) method hybridized with the Robin transmission condition (DG-RTC) is developed to solve the frequency-domain electromagnetic (EM) problems. The proposed DG method directly discretizes the vector electric field wave equation in each subdomain, and subsequently, a term named numerical flux is introduced at the subdomain interfaces to connect the solutions between neighboring subdomains. However, the numerical flux depends not only on the electric field E but also on the magnetic field H residing over the interface. Thereby, another equation is essential for solving E and H simultaneously. Realizing that H only situates at the interface of adjacent subdomains, it is thus desired that the auxiliary equation also merely involves H at the interfaces. To reach this aim, at the subdomain interfaces, a Robin transmission condition (RTC) based on the tangential continuity of EM fields across the interface is introduced as the second equation. With this proposed DG-RTC algorithm, only E is a volume variable, while H is a two-dimensional (2-D) one, and the degrees of freedom (DoFs) are thus greatly reduced compared with the traditional DG method. On the other hand, in the frequency domain, the established subdomain matrix equations are implicitly coupled with each other, resulting in a global matrix system. Directly solving it is not yet cheap. To alleviate the computational cost, a finite-element tearing and interconnecting (FETI)-like approach resorts. With the block-diagonal preconditioner and a restriction operator, the global matrix is decomposed into a surface matrix equation and several local matrix equations pertinent to each subdomain. In this way, a direct solver can be applied to solve these small matrix equations efficiently. To validate the proposed DG method in solving various EM problems, several representative examples are presented.

AB - In this work, a wave-equation-based discontinuous Galerkin (DG) method hybridized with the Robin transmission condition (DG-RTC) is developed to solve the frequency-domain electromagnetic (EM) problems. The proposed DG method directly discretizes the vector electric field wave equation in each subdomain, and subsequently, a term named numerical flux is introduced at the subdomain interfaces to connect the solutions between neighboring subdomains. However, the numerical flux depends not only on the electric field E but also on the magnetic field H residing over the interface. Thereby, another equation is essential for solving E and H simultaneously. Realizing that H only situates at the interface of adjacent subdomains, it is thus desired that the auxiliary equation also merely involves H at the interfaces. To reach this aim, at the subdomain interfaces, a Robin transmission condition (RTC) based on the tangential continuity of EM fields across the interface is introduced as the second equation. With this proposed DG-RTC algorithm, only E is a volume variable, while H is a two-dimensional (2-D) one, and the degrees of freedom (DoFs) are thus greatly reduced compared with the traditional DG method. On the other hand, in the frequency domain, the established subdomain matrix equations are implicitly coupled with each other, resulting in a global matrix system. Directly solving it is not yet cheap. To alleviate the computational cost, a finite-element tearing and interconnecting (FETI)-like approach resorts. With the block-diagonal preconditioner and a restriction operator, the global matrix is decomposed into a surface matrix equation and several local matrix equations pertinent to each subdomain. In this way, a direct solver can be applied to solve these small matrix equations efficiently. To validate the proposed DG method in solving various EM problems, several representative examples are presented.

UR - http://hdl.handle.net/10754/686362

UR - https://ieeexplore.ieee.org/document/9976312/

U2 - 10.1109/TMTT.2022.3225324

DO - 10.1109/TMTT.2022.3225324

M3 - Article

SN - 1557-9670

SP - 1

EP - 11

JO - IEEE Transactions on Microwave Theory and Techniques

JF - IEEE Transactions on Microwave Theory and Techniques

ER -

A Wave Equation-Based Hybridizable Discontinuous Galerkin–Robin Transmission Condition Algorithm for Electromagnetic Problems Analyzing

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